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Hansen’s session on teaching kids the building blocks for solving math problems at the national conference National Council of Teachers of Mathematics in late October at the Indianapolis Convention Center drew a crowded room of teachers who came to learn techniques to communicate math concepts visually with hand-held “number racks,” by having kids draw pictures to explain their answers and simply by using more precise language.

In Indiana, where new academic standards now call for students to demonstrate better mastery of math through a deeper understanding of the reasons that lead them to choose a particularly strategy to solve a problem, the ideas are especially useful.

The new standards, which detail what children must know, call for students to not just learn facts, but understand how to get answers. They learn the intuition behind borrowing in subtraction problems or figure out why an author made certain choices when writing a book. This higher level thinking and analysis helps kids be better prepared to go to college or the workforce, educators say.

Hansen, a former math teacher now with Oregon nonprofit The Math Learning Center, said it’s about time math was taught more like English, where memorization takes a back seat to understanding meaning.

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For some of her parents, who may have learned their basic math a generation ago, Natalie Merz’s second-grade math lesson might look pretty unfamiliar.

The long worksheets of stacked numbers to add, subtract, multiply or divide are gone. Students in her class at Indianapolis Public School 61 work on fewer problems at a time, working to explain how they came to their answers.

And although a math worksheet even five years ago would probably have a strict time limit — how many can you answer in one minute? — this activity had no such pressure.

But giving fewer problems and more time lets students work at their own paces and allows them develop better problem-solving skills. Rushing through timed tests, Hansen said, makes it harder for a struggling student to discover problem-solving strategies that work best for them. That can mean they actually learn less math and feel more frustration with the subject.

As she moved from group to group, Merz made gentle suggestions to her students who weren’t going far enough to explain why.

“Don’t try to do it in your head,” she told one group. “Draw a picture.”

Some students still counted on their fingers or borrowed to solve a subtraction problem and then went back to illustrate it. But most of them followed the directions: they drew a picture and wrote down the answer.

A correct “picture” next to the equation looked something like this, with tally marks visually representing numbers in the “tens” place and circles representing those in the “ones” place.

Merz was reinforcing the concept that students must recognize which numbers are “tens” and which are “ones” to fully grasp the concept of place value in addition and subtraction.

In schools strained by poverty, where children come from families with limited resources, students often struggle to articulate how they got an answer, teachers said. Teachers have to work to bridge the gap with wealthier students, where extra reading, study or academic conversation at home can help prepare kids to better explain what they mean, because the benefit of understanding how they solved a problem doesn’t just end in second grade.

“My fiance does computer programming, and he has to understand the ‘why’ logic behind what he does works,” Merz said. “There’s a process behind those jobs. I think a lot of other countries have been doing that reasoning-based math a lot more. Especially with math, it builds so much. If you don’t understand math addition, you don’t get multiplication, division, algebra or calculus.”

But when kids learn math a new way, it can make it harder for their parents not just to help them with schoolwork, but to even follow the logic themselves.

An emerging parent-child divide

At the conference, Hansen told a true story that illustrated this problem.

A father and daughter she knows were working together on the problem 17 times 99, she said. The father believed his approach was best — multiplying 17 by 99 on paper the way he learned to do it:

The daughter tried to solve it differently using a strategy called “grouping.” Her approach would break numbers down and re-group them in ways that can make the problem easy to solve quickly.

The daughter thought it was easier to multiply 100 times 17. Then, she told her dad, she would solve that equation by taking away one “group” to get the answer to the original problem.

“A group of 99?” he asked, puzzled.

“No,” she said in a matter-of-fact tone, “A group of 17.”

She calculated 17 times 100 to be 1,700 and then subtracted one group of 17 to find the correct answer: 1,683.

But her dad needed more explanation. So she drew him a picture.

The daughter illustrated her answer by drawing a grid with 17 rows and 100 columns. Altogether, the grid had 1,700 squares. When she subtracted one row of 17 squares, or taking away one group of 17 as she had said before, what was left was 1,683 boxes.

“Yeah,” he told his daughter. “I get it now.”

Visuals, Hansen said, such as number lines or grids, can help students to more quickly develop comfort with numbers and lead them to understand why a problem-solving process was used in the first place. The daughter not only knew how to get the answer, but she clearly understood the concepts behind multiplication — well enough to teach them to her father.

“Give them strategies,” Hansen told the teachers, “then drill facts.”

A different way of thinking about numbers

IPS’ curriculum team has both the new standards, and the new thinking about math, in mind when it advocates for the new strategies.

Curriculum coaches Nick Meyer and Eric Beebe believe if students learn to work through math strategies without help, it won’t just benefit them in school and in college but also better prepare them to consider the high tech jobs of the future.

“Understanding the relevance increases student engagement, but it also helps students be more successful because they can make connections from math to the everyday world,” Beebe said. “It also kind of opens the doors for them to understand what careers are associated with math and how math drives so much of what happens around us.”

But to get to there, kids have to master the basics that many adults take for granted.

Some teachers call the adult approach of doing math inside the head “mental math.” But relying on such a strategy without knowing the reasoning behind it can slow a child’s progress toward understanding.

Consider the problem 9 + 7, Hansen said. This is a problem the entire room of teachers could all do in their heads.

But when Hansen asked teachers to explain their answers, they gave a variety of different methods, but all used the same concept: grouping.

One volunteer wanted to make 9 into a 10 to make adding easier. So she split seven in two parts — a one and a six — then took the one and added it to nine. Now she had 10 and six, which add easily to make 16.

Another volunteer saw instinctively that borrowing could work the other way. She split nine in two parts — making an eight and one — then took the one to add with seven. Now she had eight plus eight, which she thought was easier to calculate to the same right answer: 16.

Hansen drew out the solutions and projected them on a screen. They looked something like this, with the arrows indicating how the broken down numbers were combined with the other to get the answer:

This strategy, at its core, shows the kind of thinking higher-level math the new standards encourage students to employ. Many adults use them instinctively. But young children must be taught how to understand numbers that way, or they are likely to resort to counting by ones or memorizing, Hansen said.

“I don’t want you to promote one-by-one counting,” Hansen told the teachers. “I want (students) to think in chunks and groups . . . (visuals) that support that one-by-one counting are the death of us.”

If the methods to solve the new problems seem complicated, it’s because they are, she said. The goal is to help kids reach a higher standard of academic reasoning. The standard algorithm — numbers stacked on top of each other with a plus or minus sign — can lead to the right answer. But it doesn’t get at the understanding behind the math — it takes more effort and thinking for students to explain why that was the best way to solve the problem.

“We could do the algorithm we learned,” Hansen said. “But we wouldn’t be able to justify. Now, standards ask students to justify that.”

Here, in a feature we call How I Teach, we ask educators who’ve been recognized for their work how they approach their jobs. You can see other pieces in this series here.

Shannon Wachowski once started a parent-teacher conference by sharing that she was concerned about the student’s lack of motivation. The boy’s mother quickly began adding criticisms of her own — alarming Wachowski enough that she started defending the teen.

It was then the student’s behavior began to make more sense to Wachowski, who teaches everything from ninth-grade earth science to college-level chemistry at Platte Valley High School in northeastern Colorado. She realized that school, not home, was the boy’s safe place.

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Wachowski is one of 20 educators who were selected to serve on the state Commissioner’s Teacher Cabinet. The group provides input to officials at the Colorado Department of Education.

She talked to Chalkbeat about how she uses parent conferences and classwork to learn students’ stories, why making Rube Goldberg contraptions boosts kids’ confidence, and what happens when she raises her hand in the middle of class.

This interview has been condensed and lightly edited.

Why did you become a teacher?
Originally a practicing chemical engineer, I became a teacher because I wanted a more fulfilling career. I had tutored chemistry in college and really enjoyed it.

What does your classroom look like?
Because my students work in teams 90 percent of the time, my tables are arranged so that students can sit in groups of four. I wrote a grant last summer for standing desks so each two person desk raises up and down. They are convenient for labs or when students need a change of scenery. My walls contain student-made license plates (an activity I do on the first day of school) and other student work from class, including various Chemistry Cat memes!

Fill in the blank. I couldn’t teach without my ________. Why?
My heart. Initially I became a teacher because I loved my content. I soon realized however, that while content is important, developing relationships with students is paramount. No learning will happen if positive relationships are not established first. When I am frustrated with student behavior, I try to put myself in their place and respond in a caring and compassionate manner.

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What is one of your favorite lessons to teach?
One of my favorite lessons is when my students build Rube Goldberg devices. It gets somewhat chaotic because they are working in teams and materials are everywhere, but every single student is engaged. In the end, they can apply what they know about energy to design a multi-step contraption. I have seen very low-confidence students excel at this activity, and it is very rewarding to see them experience success in a science class.

How do you get your class’s attention if students are talking or off task?
One strategy I’ve recently started using came from my experience leading professional development for other teachers. I will be somewhere in the middle of the room (usually not the front) and raise my hand. When students see me raise my hand, they will raise theirs and pause their conversation. Then other students see those students and raise their hand, etc. Once everyone is quiet, then I’ll make my announcement. Like all other strategies, I need to practice being consistent with it.

How do you get to know your students and build relationships with them? What questions do you ask or what actions do you take?
I always plan the first couple of days for “get to know you” activities. My students design their own paper license plates using whatever letters, numbers, or design they would like. They then have 30 seconds to talk about their license plates.
I noticed that in some of my more challenging classes I needed a way to better connect with my students. At the beginning of most class periods I share some sort of funny story about what happened to me the evening prior — for some reason, I am never short of these stories — or a picture of my dog, or my latest climbing adventure. Sharing this information does not take long and eventually, students will ask if I have a story to share if I haven’t done so in a while. This also leads to them sharing stories with me, and finding that we may have more in common than we think.

Tell us about a memorable time-good or bad-when contact with a student’s family changed your perspective or approach.
At parent-teacher conferences one year I had a parent come in with their student. This student was not the most motivated individual — not disrespectful, just did not seem to want to do anything with his time. As I was explaining this to his parent, the parent started talking very negatively to and about the student, so much so that I found myself trying to defend the student and bring up positive qualities about his character. This interaction helped me to understand some of the student’s behavior in class, as well as realize that for some students, school is their safe place. There are often lots of reasons for a student’s behavior that I may not be aware of, which is why it is important to get to know each student and their situation as best as possible.

What are you reading for enjoyment?
When I have time outside of school, one of the things I enjoy doing is throwing pottery. I am currently reading “Science for Potters” by Linda Bloomfield. It combines my love of science and art into one book.

What is the best advice you ever received?
Since I teach a variety of levels, I often have one class that challenges my classroom management skills. This can be frustrating as I am the type of person that would like to achieve perfection in every circumstance. When I have a discipline issue in my class, I often see it as a personal failure. My husband often reminds me that “You can’t control other people’s behavior, you can only control your response to it.”

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Jean-Baptiste picks up where she leaves off: “We need change in this time of doom. It shouldn’t be the case that we’re losing lives too soon. I shouldn’t feel afraid inside my school. We need change.”

We checked in with Kyle Fackrell, Lavelle Prep’s longtime music teacher, who has worked with Jean-Baptiste, Soler, and their classmates for nearly five years, since their introductory eighth-grade music class. Here’s what he told us about the song, his students, and their ambitions.

How the song came to be: “I knew that my students were very passionate about this subject. When I learned about the walkout coming up and that it would be coming up soon, I was aware of these students and their songwriting abilities, and I suggested the idea of writing a song. They really just ran with it.”

What the process was like: “We’ve worked together a lot and have made a lot of music together. When I proposed this idea it was like clockwork. It was really exciting to see how fast Jerramiah could come up with the ideas.”

On the students’ goals: “We just wanted to help the movement. I was having that conversation with my students today, should the song get the success we hope it gets, that would be great, but really want we to maintain our genuine interest in making a difference with the song. I’m just supporting them.”

What the reaction has been: “It’s been very positive. … Everyone who hears the song is blown away. It really is thanks to the talent of the young students that I’m blessed to be helping them develop.”

On what motivates his students: “None of them were coming at it from knowing people who were in a shooting. They’re just very aware and intelligent students. I think the point that the students in Florida are making is that a lot of people underestimate kids and youth, and I think these students are also underestimated — about how much they are aware of what’s going on in the world, and that they should have a say.”